6 Major overhaul

## 1Examples,smallest$n$

As ulrich points out, there exist abelian surfaces in $\mathbb P^4$, so those give you $c=2$ with $3$-dimensional singularities, that is, $n=5=2c+1$. See below how to get examples for any $n\geq 5$. For $c=2$ this is the answersmallest $n$ you can get. Here is why: For $n\leq 4$ anything of codimension $c=2$ would be of dimension at most $2$ and hence if it is normal, it is $S_2$ and in particular CM.

Now

## 2Non-indecomposables

(Note: this is here, if because I originally did not read the requirements carefully. Then I didn't feel like erasing it. )

If you do did not need required an isolatedindecomposable singularity, then you can could take the product of this $X\subset \mathbb A^{2c-1}$ and an arbitrary $\mathbb A^r$. Of course, so this is why you made that requirement. Anyway, you'd get

If you wanted isolated indecomposable singularities, I would expect that the codimension is actually increasing with the dimension. In other words, I would expect low codimensional examples in low dimension and not in (arbitrarily) high dimension.

Finally

## 3 Indecomposable vs. isolated

The previous point shows why Hailong assumed indecomposable. I claim that one may as well also assume isolated. At least to start.

First, let $X$ be an example as required, of dimension $d=\dim X$ and of codimension $c$. Let $s=\mathrm{Sing} X$ the dimension of the singular set of $X$ and assume that $\mathrm{Sing} X$ is irreducible. Take a general complete intersection of codimension $s$. This will have dimension $d-s$, codimension $c$ and it is an isolated non-CM log canonical singularity.

Next, let $X$ be a codimension $c$ and it is an isolated non-CM log canonical singularity and consider a $\mathbb Q$-Gorenstein deformation of $X$ over a base of dimension $s$. (A $\mathbb Q$-Gorenstein deformation means that the relative dualizing sheaf of the family is a $\mathbb Q$-line bundle and its line bundle powers restrict to the appropriate power of the dualizing sheaf of the members of the family).The total space of the deformation will be an example of the kind you want. (For an indecomposable example you need a deformation without a trivial component).

The fact that this gives you an example that you want is non-trivial. It is log canonical by inversion of adjunction see the main result of Kawakita's paper and it is non-CM at every point of the singular set by Corollary 1.3 of this paper.

From this you can conclude that any example you get will be a deformation of an isolated example. A priori you get that at the generic points of the singular set. At non-generic points those are still degenerations (i.e., non-small deformations) of the general ones.

The problem you run into is that you need non-trivial deformations of these singularities and they have finite dimensional versal deformation spaces, so you can't get too far with this idea. (Angelo will correct me if this is wrong, since he is one of the ultimate experts on this. See also Artin's extended work on this topic.)

This suggests that given your restriction of being indecomposable, regarding your "every big enough $n$" question, it seems that in order for that to happen you really need low codimensional isolated examples, which will be hard to construct since you can't get too far with cones (cf. ulrich's and Angelo's comments).

On the other hand we do get new examples out of this: A flat family of polarized abelian/CY/etc varieties will give you a flat family of the cones over them. As long as the family has maximal variation (i.e., the moduli map of the base is generically finite) the resulting singularity will be what you want. So, this way you can get higher dimensional examples, but it seems that all constructions are limited in dimension.

All of this suggests that the answer to your big enough question will be "no".

## 4 Better examples (manageability over low codimension)

To get an example that you want, you do not need to have an abelian variety for this the cone construction. If $A$ is a smooth projective variety of dimension $d$ such that $\omega_A\simeq \mathscr O_A$ and there exist two integers $i,m\in\mathbb Z$ such that $0<i<d$ and $$H^i(A,\mathscr O_A(m))\neq 0,$$ then the cone over $A$ has non-CM log canonical singularities. In other words, to beat the above given bounds get more examples you just need to find such subvarieties of low codimension.

Unfortunately, the

The obvious embedding via Segre gives larger codimension than what you get from the above procedure, so this does not obviously give you smaller codimensional examples, but they may be more manageable since you have the product of a hypersurface and a plane curve and even if the codimension is high, you know pretty well the embedding pretty well. So, perhaps from a practical point of view these are even perhaps better examples after all.

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Let $A$ be an abelian variety and $X$ a cone over $A$. Then $X$ is log canonical, but as soon as $\dim A\geq 2$, then $X$ is not Cohen-Macaulay. (This you can see by computing the local cohomology at the vertex).

For the $c=1$ case: those are obviously CM. Otherwise one can study the codimensions of various embeddings of abelian varieties.

As ulrich points out, there exist abelian surfaces in $\mathbb P^4$, so those give you $c=2$ with $3$-dimensional singularities, that is, $n=5=2c+1$. See below how to get examples for any $n\geq 5$. For $c=2$ this is the answer. Here is why: For $n\leq 4$ anything of codimension $c=2$ would be of dimension at most $2$ and hence if it is normal, it is $S_2$ and in particular CM.

So, let's assume that $c>2$.

As any quasi-projective variety $A$ of dimension $d$ maybe embedded in $\mathbb P^{2d+1}$ (Embed in some $\mathbb P^N$ and notice that the closure of the secant variety of $A$ is of dimension $2d+1$, so as long as $N>2d+1$, one may find a point and a projection that gives an embedding of $X$ into $\mathbb P^{N-1}$. Repeat.), you can do that with abelian varieties as well. This gives you $c=d+1$, so turning it around, for any $c>2$ you can find an $X$ that you're looking for in $\mathbb A^n$ with $n=2c-1$.

Now, if you do not need an isolated singularity, then you can take the product of this $X\subset \mathbb A^{2c-1}$ and an arbitrary $\mathbb A^r$, so you get $$X_{c,r}:=X\times \mathbb A^r\subset \mathbb A^{2c-1+r}$$ of codimension $c$ with $n=2c-1+r$. In other words, yes, you can construct such an $X$ for all $n$ big enough. I am not sure whether the bound $2c-1$ is optimal, but I have a feeling that you can't get much better than that. Note that this construction works for $c=2$ as well, so you can get examples in all dimensions $n\geq 5=2\cdot 2+1$.

If you wanted isolated singularities, I would expect that the codimension is actually increasing with the dimension. In other words, I would expect low codimensional examples in low dimension and not in (arbitrarily) high dimension.

Finally, you do not need to have an abelian variety for this construction. If $A$ is a smooth projective variety of dimension $d$ such that $\omega_A\simeq \mathscr O_A$ and there exist two integers $i,m\in\mathbb Z$ such that $0<i<d$ and $$H^i(A,\mathscr O_A(m))\neq 0,$$ then the cone over $A$ has non-CM log canonical singularities. In other words, to beat the above given bounds you just need to find such subvarieties.

Of course, complete intersections do not satisfy this, but for example the product of any two CYs do. So, you could take, say, a CY hypersurface $H$ and an elliptic curve $E$ and then $A=H\times E$ satisfies to condition (and $A$ is generally neither CY nor abelian).

Unfortunately, the obvious embedding via Segre gives larger codimension than what you get from the above procedure, so this does not obviously give you smaller codimensional examples, but they may be more manageable since you have the product of a hypersurface and a plane curve and even if the codimension is high, you know pretty well the embedding. So, perhaps these are even better examples after all.

Let $A$ be an abelian variety and $X$ a cone over $A$. Then $X$ is log canonical, but as soon as $\dim A\geq 2$, then $X$ is not Cohen-Macaulay. (This you can see by computing the local cohomology at the vertex).

For the $c=1$ case: those are obviously CM. Otherwise one can study the codimensions of various embeddings of abelian varieties.

As ulrich points out, there exist abelian surfaces in $\mathbb P^4$, so those give you $c=2$ with $3$-dimensional singularities, that is, $n=5=2c+1$. See below how to get examples for any $n\geq 5$. For $c=2$ this is the answer. Here is why: For $n\leq 4$ anything of codimension $c=2$ would be of dimension at most $2$ and hence if it is normal, it is $S_2$ and in particular CM.

So, let's assume that $c>2$.

As any quasi-projective variety $A$ of dimension $d$ maybe embedded in $\mathbb P^{2d+1}$ (Embed in some $\mathbb P^N$ and notice that the closure of the secant variety of $A$ is of dimension $2d+1$, so as long as $N>2d+1$, one may find a point and a projection that gives an embedding of $X$ into $\mathbb P^{N-1}$. Repeat.), you can do that with abelian varieties as well. This gives you $c=d+1$, so turning it around, for any $c>2$ you can find an $X$ that you're looking for in $\mathbb A^n$ with $n=2c-1$.

Now, if you do not need an isolated singularity, then you can take the product of this $X\subset \mathbb A^{2c-1}$ and an arbitrary $\mathbb A^r$, so you get $$X_{c,r}:=X\times \mathbb A^r\subset \mathbb A^{2c-1+r}$$ of codimension $c$ with $n=2c-1+r$. In other words, yes, you can construct such an $X$ for all $n$ big enough. I am not sure whether the bound $2c-1$ is optimal, but I have a feeling that you can't get much better than that. Note that this construction works for $c=2$ as well, so you can get examples in all dimensions $n\geq 5=2\cdot 2+1$.

If you wanted isolated singularities, I would expect that the codimension is actually increasing with the dimension. In other words, I would expect low codimensional examples in low dimension and not in (arbitrarily) high dimension.

Finally, you do not need to have an abelian variety for this construction. If $A$ is a smooth projective variety of dimension $d$ such that $\omega_A\simeq \mathscr O_A$ and there exist two integers $i,m\in\mathbb Z$ such that $0<i<d$ and $$H^i(A,\mathscr O_A(m))\neq 0,$$ then the cone over $A$ has non-CM log canonical singularities. In other words, to beat the above given bounds you just need to find such subvarieties.

Of course, complete intersections do not satisfy this, but for example the product of any two CYs do. So, you could take, say, a CY hypersurface $H$ and an elliptic curve $E$ and then $A=H\times E$ satisfies to condition (and $A$ is generally neither CY nor abelian).

Unfortunately, the obvious embedding via Segre gives larger codimension than what you get from the above procedure, so this does not obviously give you smaller codimensional examples, but they may be more manageable since you have the product of a hypersurface and a plane curve and even if the codimension is high, you know pretty well the embedding. So, perhaps these are even better examples after all.

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